An important generalization of Galois extensions of fields is to Hopf-Galois extensions of associative rings, which Schneider proved can be characterized in terms faithfully flat Grothendieck descent. I will begin by recalling this classical theory and then sketch recent homotopical generalizations, motivated by Rognes' theory of Hopf-Galois extensions of structured ring spectra. In particular, I will present a homotopical version of Schneider's theorem, which describes the close relationships among the notions of Hopf-Galois extensions, Grothendieck descent, and Koszul duality within the framework of Quillen model categories.